15.6: Conclusion

You have worked hard trying to understand these proofs of soundness and completeness. I too have worked hard, first in understanding them and then in my efforts to write them up in a clear and accessible form. Working on the strength of the presentations of others, I will be very happy if I have made some small contribution to improving the accessibility of soundness and completeness and if I have avoided both horns of the dilemma of too much complication versus inaccuracies in the proofs. Whatever I have accomplished, I am sure that my presentation can be improved. I welcome your comments and suggestions. In the meantime, you should not be discouraged if you have found part I1 of this text to be very difficult. Soundness and completeness are substantial mathematical results. If you understand them only in a fragmentary way, you can greatly improve your grasp by patiently going over these chapters again.

CHAPTER CONCEPTS

In reviewing this chapter, be sure you have a firm grasp on the following:

Interpretation

Unnamed Object

s-Variant Truth of an Existentially Quantified Sentenced)

Truth of a Universally Quantified Sentence

Infinite Tree

Infinite Semantic Tableau Derivation

Downward correctness of a Truth Tree Rule

Upward correctness of a Truth Tree Rule

Interpretation Represented by an Infinite Truth Tree Branch

Restrictions on the Derivation Rule for VI

Restrictions on the Derivation Rule for 3E

Terminal Interpretation in a Semantic Tableau Derivation

Closed Semantic Tableau Derivation

Compactness for Predicate Logic

Interpretation for a Language with Identity

Interpretation for a Language with Function Symbols

Contributors

Paul Teller (UC Davis). The Primer was published in 1989 by Prentice Hall, since acquired by Pearson Education. Pearson Education has allowed the Primer to go out of print and returned the copyright to Professor Teller who is happy to make it available without charge for instructional and educational use.

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